The fact that we consider complex-valued “para-Hermitian” structures implies that,
unlike the usual literature in DFT and related matters, we can analyse different signatures of the metric,
by imposing different reality conditions.
Since we work in a 4-manifold \(M\)  (which we assume to be orientable), this means that we will deal with
Lorentz, Riemannian (or Euclidean) and split (or neutral) signature.
The Lorentzian case is relevant, of course, for general relativity (and in particular for
the so-called hyper-heavenly spaces [1]});
the Riemannian case is relevant because e.g. of the Atiyah-Hitchin-Singer approach to twistor theory [2]};
and the split case can be related, since the work of Ooguri and Vafa [3]},
to the geometry of strings with \(N=2\)  supersymmetry. (It is also related to the LeBrun-Mason twistor construction
in split signature [4]}.)
